A sketch of the temperature profiles near the plates is displayed in figure REF , where we also show how the thermal BL thicknesses are defined. Near the cold top plate (see figure REF a), the thermal BL thickness \(\lambda _{cold}\)  is defined through the usual convenient definition via the slope of the temperature profile at the plate from assuming a pure conductive thermal BL and a well-mixed bulk [1]}. As \(\lambda _{cold}\)  we take that distance from the plate, where the tangent to the temperature profile at the plate reaches the mean centre temperature \(\bar{\theta }\) .
However, this definition cannot be directly applied near the hot bottom plate, due to the bubbles. As the bubbles reduce the area occupied by the liquid, the conducting area varies with \(z\) . We therefore correct the solution for pure thermal conduction in the hot bottom BL based on heat flux \(Q\)  conservation,
\(Q=k_l\frac{\partial \theta }{\partial z}[1-S(z,S_0,\tilde{h},n)]=k_l\frac{\bar{\theta }H}{\lambda _{cold}}.\) 
